Testing Linear Pricing Algorithms for use in Ascending ... - CiteSeerX

Testing Linear Pricing Algorithms for use in Ascending Combinatorial Auctions1, 2 Melissa Dunford‡, Karla Hoffman†, Dinesh Menon‡, Rudy Sultana‡ and Thomas Wilson‡ †

Department of Systems Engineering and Operations Research, George Mason University, Fairfax, VA 22030 ([email protected]) ‡

Optimization Solutions Group,

Decisive Analytics Corporation, Arlington, VA 22202 Contact: ([email protected]) Abstract In a one-sided ascending combinatorial auction, bidders place bids on packages of items. When values are interdependent, multi-round auctions are employed to allow price discovery. In such designs, bidders desire feedback on the estimated prices of items and packages in each round. In addition, prices are often used to set the minimum acceptable bid amounts for the next round. Due to the combinatorial nature of the winner determination problem, determining the prices of the individual items is not straightforward. The ascending proxy auction (Ausubel-Milgrom, 2001) has been proposed as a design, where the minimum acceptable bid amounts are a very small increment above the previous high bid on an item or package. These translate to nonlinear and non-anonymous (bidder specific) pricing. When agents are substitutes and the buyer sub-modularity condition is satisfied, the ascending proxy design achieves an efficient outcome at minimum competitive equilibrium prices. But the requirement for an infinitesimally small increment size, results in an auction with a very large number of rounds. In this paper, we propose two alternative linearpricing schemes that use the game-theoretic concept of the nucleolus to provide price estimates for each item and to determine minimum acceptable bid amounts. We study the effects of such pricing on the length of the auction, the efficiency of the outcome and the prices paid by the winning bidders. We use the linear pricing schemes in an ascending proxy auction framework and benchmark the results against an ascending proxy auction and a faster-to-compute variant of this algorithm that we label the “accelerated proxy auction design”. We also compare our results with those obtained using previously proposed linear pricing schemes.

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This research was partially funded by the Federal Communications Commission under a contract to CompuTech, Inc. All views presented in this research are those of the authors and do not necessarily represent the views of the Federal Communications Commission or its staff. 2 The authors are grateful to David Johnson, Evan Kwerel, and David Parkes, for their helpful suggestions and comments. Of course, all errors are our own.

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1

Introduction

Combinatorial auctions are auctions in which multiple items are concurrently sold and where bidders are allowed to declare a single value for a collection of any subset of the items to be auctioned. The flexibility of such declarations can increase the efficiency of the auction while reducing the risk to bidders of receiving only some subset of the items that they require. However, combinatorial auctions require that the auctioneer be able to determine the optimal winners by solving a combinatorial optimization problem. In addition, if choosing an ascending multi-round auction design in order to allow price discovery during the auction, the auctioneer must also provide information about the current value of each package. This information is used for two related purposes: (1) to specify the minimum bid for each item or package in the next round and (2) to provide valuation information to bidders so that they can determine what might be required for a bid to be “winning” in a subsequent round. While pricing information is easy to ascertain in simultaneous multi-round auctions without package bidding, (i.e. where bids can be placed on only single items), pricing information for combinatorial auctions is not well defined. Bidders provide only aggregate package prices without providing the information about how each of the individual components that made up the bundle contributed to the overall price. Attempting to disaggregate these bundles into single item prices unambiguously is not possible. Also, since there are many ways that some bundle might “partner” with other packages to create a “winning” set, determining the minimal cost partnering for a given package by a given bidder is a complex problem. To further complicate the pricing issue, bidders may view certain items as substitutes and other items as complements. In the case where items are substitutes, bidders are likely to express sub-additive values for their packages. That is, the value of a package of items is less than or equal to the sum of the values of the items that make up the package. In the complementary case, bidders are likely to express super-additive values for packages. In this case, the value of a package of items is greater than or equal to the sum of the values of the items that make up the package. When items can be both substitutes and complements for bidders, providing unambiguous, complete and accurate price information is an unsolved problem. The combinatorial auction literature has proposed a variety of alternative pricing algorithms based on optimization theory. We will show the amount of information needed to obtain exact dual pricing information based on the winner determination problem is computational intractable, and we then present a variety of approximate pricing strategies that are tractable. We acknowledge that different pricing approximations may be more appropriate depending on whether most bids have superadditive, additive or sub-additive valuations. We will touch on the pros and cons of each algorithm, and then describe an experimental test-design to evaluate some of the most promising pricing alternatives. The experimental testing is based on a simulation program that models bidders as “intelligent agents”. Each computer agent has been

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provided with specific goals and valuations on both packages and individual items. Because this simulation is modeled after FCC spectrum auctions, we assume that each item being auctioned is a unique entity. In this case, an item is a license to use a specific bandwidth of frequency in a given geographic region, and adjacency is defined either geographically or as bandwidth adjacency. In order ascertain the quality of the prices obtained from the smoothed anchoring method, we tested the linear pricing algorithms and measured the quality of the prices obtained from each of the algorithms in terms of the auction length, the efficiency of the final outcome and the prices paid by the winning bidders. The prices were benchmarked against those obtained by an accelerated version of the ascending proxy auction. This paper presents the results of this testing. Section 2 provides background on duality theory and presents the pricing issues related to combinatorial auctions. Section 3 contains a description of the linear pricing algorithms tested. Section 4 describes the test environment and the test design as well as provides the test cases and results. Finally, Section 5 provides analysis and conclusions of the test results.

2 Winner Determination Problem, Duality Theory and Pricing Issues The winner determination problem for a combinatorial auction with unique entities is a linear integer optimization problem where the constraints require that each item be awarded at most once and where the objective function is to maximize the revenue obtained from the auction. We first discuss this simplest case and then discuss the impact of adding additional constraints, such as mutual exclusivity of bids. Every linear optimization problem has an equivalent dual linear optimization problem whereby the roles of the variables and the constraints are reversed. That is, for every variable in the original problem there is a constraint in the dual problem and vise-versa. The winner-determination problem has a linear objective function and linear constraints. Thus, the values of the corresponding dual variables provide pricing information associated with each item and measure the monetary cost of not awarding the item to whom it has been provisionally assigned. This information could be automatically provided with no additional computation if the problem were a linear optimization problem. However, the winner-determination problem has one added set of constraints – items are not divisible, i.e. the entire item is awarded to one and only one bidder, or it is not awarded to any bidder. Thus, the winner determination problem becomes: [WDP] : max ∑ b j x j j ∈B

Subject to :

∑x

≤ 1,

∀

i∈I

x j ∈ {0,1},

∀

j∈B

j

j ∈B i

where: 3

I is the set of items being auctioned B is the set of all considered bids, Bi is the set of considered bids containing item i, and bj is the bid amount on bid j These additional integrality constraints transform the problem from a convex optimization problem to a non-convex problem. In this more general case, specifically when items show strong complementarities, the linear dual problem might have an objective function value that is strictly greater than the objective function of the primal problem. In this case, the dual prices will overestimate the true values of the items. In this non-linear case, strong duality requires the following two properties: (a) the primal and dual objective functions must be equal and (b) complementary slackness must be satisfied (i.e. a dual price can only be positive when the item is awarded). Under the conditions of integrality, a linear pricing function satisfying strong duality conditions might not exist. Wolsey (1981) examines if there is a nonlinear function that satisfies the properties of strong duality – namely that there exists a nonlinear dual price function that satisfies the complementary condition and for which the primal and dual solution values are equal. He proves that such a price function must be both non-linear and super additive. We note that super-additive functions violate the idea of substitutability. The winnerdetermination problem will naturally choose a super-additive bid over the items that make up that bid. As an example, consider a bidder j that provides bids on item A, item B, and the package AB. If the bid amounts on these bids are such that bj(A) + bj(B) ≤ bj(AB) (where bj(A) represents the bid amount placed on bid A by bidder j), then the optimization will correctly choose the bid AB over the bidders two separate bids A and B. On the other hand, if the goods are substitutes, i.e. bj(A) + bj(B) ≥ bj(AB) then, the optimization may erroneously produce a revenue equal to bj(A) + bj(B) rather than bj(AB) because it chose to put both items A and B into the solution when the bidder specified a lower value bj(AB), if both were won. Allowing the bidder the opportunity to express “exclusive-OR” relationships can eliminate this problem. Thus, if a bidder wants to express a sub-additive relationship between items A and B, he must do so by stating an “exclusive-OR” relationship among these packages. For our example above, a bidder will provide bid amounts for the bids: A, B, and AB but will specify that the solution can choose only one of these three bids. A simple way of allowing this added expressiveness is to have the bidder create packages using a “dummy” or “phantom” good (call it D) [Fujishima, et al, 1999]. Thus, the bidder creates packages AD, BD, and ABD. Since good “D” can only be awarded once, the exclusivity of these three bids is assured. With the addition of dummy goods to express any sub-additive relationships, the winner determination problem will correctly solve the problem. Here we are assuming that the formulation has added a collection of dummy items to the set of the original items. Thus, an additional set of constraints to the winner-determination problem will specify which bids of a given bidder are to be treated as mutually exclusive. These additional constraints provide price information that is unique to the bidder and explains why we might wish to have pricing that is bidder-specific.

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Bikhchandani and Ostroy (1998) reformulate the winner determination problem in an exponential number of variables where the solution to the resulting integer problem is guaranteed to be solvable with linear programming. The reformulation requires that one introduce a variable for every feasible integer solution, i.e. each variable is a collection of bids that together assign each item exactly once and allow only one bid from a given bidder2. Thus, the optimization must only decide which column to choose as the optimal solution. Additional constraints are added to the problem that are redundant but allow pricing information that provides a dual price for every package of every bidder. They do this by adding constraints that track the effect of removing or adding a bidder. In this way the linear program supplies each agent’s marginal product, i.e. it measures the effect of removing agent j from the problem. Since this formulation requires explicitly generating every feasible solution to the integer problem, the formulation is not a practical approach for obtaining pricing information to large combinatorial auctions. However, by looking at such formulations, the authors provide insights into what is required to obtain proper dual prices. The authors examine the dual prices that are produced by the solution to the extended formulation, where each feasible solution is a variable, and show that the dual prices provided by this formulation will only be VCG prices when “agents-are-substitutes” (i.e. a condition that requires that the marginal contribution of a group of agents must be at least equal to the sum of the marginal contributions of the individual members of the group). They also show that even when “agents-are-substitutes”, the resulting pricing provided by the dual optimization problem is both nonlinear and non-anonymous, i.e. the agent’s payments must be broken into a non-anonymous (i.e. bidder specific) component and a non-linear component associated with the given package price. They also show that the linear programming relaxation to the original winnerdetermination problem will only be integer optimal if (1) each agent’s value function satisfies the free disposal property (the value of a package A is less than or equal to the value of any larger package that contains A) and (2) the buyer submodularity property holds (a stronger condition than “agents-are-substitutes”3). When complementarities are present, the optimization is likely to choose bids that include these synergistic values. The associated super-additive dual prices will then not be VCG prices. In this case, if a linear pricing scheme is used, a bidder does not necessarily have the incentive to bid sincerely since there does not exist pricing that will guarantee that he receives all of the extra gains that his participation confers. For more on the relationship between linear programming and Vickrey Auctions, we refer the reader to Bikhchandani, et al (2001).

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The formulation of Bikhchandani and Ostroy assumes that each bidder expresses a bid on every combination of items that he is interested in obtaining. Thus, they assume that every bid of a bidder is mutually exclusive. 3 For definitions of “agents-are-substitutes” property and buyer submodularity, see Section 4 of this paper.

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When preferences are neither consistently sub-additive (substitutes) nor super-additive (complements), obtaining such nonlinear, non-anonymous pricing requires the determination of the best coalition for every non-winning bid. Because enumerating all such coalitions is computationally prohibitive, we have considered alternative approximations to the true prices. Specifically, we acknowledge that there will be bidders who view items as substitutes and others who view items as complements. A compromise must be arrived at that does not bias consistently against either type of bidder. Although linear pricing cannot accommodate all aspects of the pricing associated with the non-linear, non-convex, winner determination problem, there are still good reasons for considering its use for determining future bid requirements. Firstly, even “perfect” pricing is only correct when all other aspects of the problem remain fixed – i.e. when bid amounts remain the same on all other bids and when no new bids are submitted. Secondly, a dual price associated with a given constraint is only “correct” when one changes this single restriction (the right-hand-side of this constraint) by a very small amount. In our case, the item is either won or it is not. Changes to a constraint would either remove the item entirely from consideration or create a second identical item. Both of these changes can be dramatic to the system, and dual pricing is unlikely to be valid for such dramatic changes. Thus, even non-linear, non-anonymous pricing has serious limitations in the context of the winner determination problem. Additionally, in an ascending bid auction, bidders need pricing information that is easy to use and understand, and is perceived to be “fair”. We mean “easy to use” in the sense that bidders can quickly compute the price of any package – whether or not it had been previously bid. Often, bidders want to know what it would take for such a bid to be “competitive”, i.e. have some possibility of winning in the next round. Bidders may all perceive such prices to be “fair” since all bidders must act on the same information. Finally, linear prices are likely to move the auction along and deter such gaming strategies as “parking” (choosing to bid on packages that currently have very low prices knowing that such packages cannot win). We describe the linear pricing algorithm used by the FCC as well as a number of alternative linear pricing algorithms that can be used to provide pricing information in an ascending auction. Since these algorithms involve solving one or more linear programs, they are easy to compute. All these algorithms are based on the dual of the winnerdetermination problem. We compare these to an approach that uses non-anonymous, non-linear pricing – the Ausubel-Milgrom ascending proxy auction design.

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Linear Pricing Algorithms

In this section, we review the linear relaxation of the winner determination problem and its associated dual. Next the idea of pseudo-dual prices is presented before summarizing a number of alternative linear pricing approximations, including the smoothed anchoring method used by the FCC. Finally, we suggest two new ideas.

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3.1 Linear Relaxation of the Winner Determination Problem The linear relaxation of the winner determination problem, denoted [WDP LP] for a collection of heterogeneous items is: [WDP LP ] : max ∑ b j x j j ∈B

Subject to :

∑x

j

≤ 1, ∀ i ∈ I

j ∈B i

0 ≤ x j ≤ 1, ∀ j ∈ B

where: I is the set of items being auctioned B is the set of all considered bids, Bi is the set of considered bids containing item i, and bj is the bid amount on bid j The considered bids, B, include minimum opening bids on all individual items by the auctioneer. For each item i∈ I, we assume that the price must be at least the minimum opening bid price, ri. The dual of (WDP LP) is: [WDP LP Dual ] : min ∑ π i i∈L

Subject to :

∑π

i

≥ bj ,

∀

j∈B

i∈I j

π i ≥ ri ,

∀ i∈I

where, I j is the set of items contained in bid j. [WDP LP Dual] provides pricing information associated with each item, measuring the monetary cost of not awarding the item to whom it has been assigned. 3.2 Pseudo-Dual Prices If the linear optimization relaxation to the winner-determination problem was integer optimal than the resulting prices, πi, would satisfy the inequalities, ∑ π i ≥ b, ∀ j ∈ B

and

∑π =b , ∀ j ∈W , where W is the set of winning bids. i

j

i∈I j

7

i∈I j

Since such prices might not

exist one approach is to use pseudo-dual prices4 rather than the dual prices. These pseudo-dual prices are obtained by forcing the sum of the dual prices of the items comprising a provisionally winning bid to equal its respective bid amount but allowing the prices of non-winning bids to be less than the maximum price bid for that package. For example, suppose there are two bids in the provisionally winning set in round t: a bid on item A for $10 and a bid on package BC for $25. The pseudo-dual price of A will equal $10 and the sum of the pseudo-dual prices of B and C will equal $25. These restrictions ensure that the sum of the pseudo-dual prices equals the maximum revenue for the round (e.g. $35) and that minimum acceptable bid amounts reflect the bid amounts of bids in the provisionally winning set. In order to ensure these restrictions, we allow prices of non-winning packages to be less than the maximum bid amount on that package. Thus, the pseudo-dual prices for each item i, denoted by πi are required to satisfy the following constraints:

∑π

i

+ δ j ≥ bj ,

∀ j∈B \W

(1)

∑π

i

= bw ,

∀ j ∈W

(2)

δ j ≥ 0,

∀ j∈B \W

(3)

π i ≥ ri

∀ i∈I

(4)

i∈I j

i∈I j

where, W⊂ B is the provisionally winning bid set and δ j is a slack variable that represents the difference between the bid amount of non-winning bid j and the sum of pseudo-dual prices of the items contained in non-winning bid j By keeping constraints (1)-(4), we have considerable flexibility in choosing an objective function that will help in selecting among multiple solutions while still ensuring that the sum of the pseudo-dual prices yields the maximum revenue of the round. There are likely to be many solutions (i.e. many sets of dual prices) that satisfy this constraint set. For instance, in the example provided earlier, the pseudo-dual prices of B and C might be any two numbers that together sum to $25. Since we want the solution to be as close to integer optimality as possible, we want to minimize the total infeasibility, i.e. we wish to minimize ∑ δ i . Pseudo-dual prices for each item i, denoted πi, can be obtained by j∈B \W

solving the following linear program:

4 In our research we found this term first applied to auction pricing in the paper by Rassenti, Smith and Bulfin (1982).

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[CP ] : zδ* = min

∑

j ∈B \ W

δi

Subject to :

∑π

i

+ δ j ≥ bj ,

∀ j∈B \W

(1)

∑π

i

= bj ,

∀ j ∈W

(2)

δ j ≥ 0,

∀ j∈B \W

(3)

π i ≥ ri ,

∀ i∈I

( 4)

i∈I j

i∈I j

The solution to this problem is not necessarily unique. In fact, testing has shown that using [CP] in an iterative auction can result in significant changes in the pseudo-dual price of an item from round to round. Although the prices of items should be allowed to reflect real change (both increases and decreases) in the way bidders value the items over time. Large oscillations in minimum acceptable bid amounts for the same bid that are due to factors unrelated to bidder activity, such as multiple optimal primal or dual solutions, can be confusing to bidders and may encourage unwanted gaming of the system. We therefore solve a second optimization problem that chooses a solution in a way that reduces the magnitude of price fluctuations between rounds. This method is known as smoothed anchoring since the method anchors on exponentially smoothed prices from the previous round when determining prices for the current round. We present the details of the smoothed anchoring method next. 3.2.1 The Smoothed Anchoring Method The smoothed anchoring method involves solving two optimization problems. First, [CP] is solved to obtain the minimum sum of slack. Second, a linear quadratic program is solved with an objective function that applies the concepts of exponential smoothing to choose among alternative pseudo-dual prices with the additional constraint on the problem that the sum of the slack variables equals zδ* (the optimal value of [CP]). This objective function minimizes the sum of the squared distances of the resulting pseudodual prices in round t from their respective smoothed prices in round t-1. Let π it be the pseudo-dual price of item i in round t. The smoothed price for item i in round t, is calculated using the following exponential smoothing formula:

pit = απ it + (1 − α ) pit −1 where, pit −1 is the smoothed price in round t-1, 0 ≤ α ≤ 1, and pi0 = the minimum opening bid amount for item i.

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The following quadratic program (QP) will find the pseudo-dual price, π it for each item i in round t: that minimizes the sum of the squared distances from the respective smoothed prices in round t-1 while assuring that the pseudo-dual prices sum up to the provisionally winning bid amounts.

[QP ] : min

∑ (π i∈L

t i

− pit −1 )2

Subject to :

∑π

t i

∑π

t i

i ∈I

+ δ j ≥ bj ,

∀ j ∈ Bt \ W t

(1)

∀ j ∈W t

(2)

j

= bj ,

i ∈I j

∑

δ j = zδ*

(3)

j ∈B t \ W t

δ j ≥ 0,

∀ j ∈ Bt \ W t

π ≥ ri ,

∀ i ∈I

t i

( 4) ( 5)

Note that problem (QP) has the same constraints as [CP], but has added the additional restriction (3) that the sum of the δ j ’s is fixed to the value zδ* , the optimal value from [CP]. Among alternative prices that satisfy all constraints, the objective function of this optimization problem chooses one that forces the pseudo-dual prices to be as close as possible to the previous round’s smoothed price. Thus, this method is called the smoothed anchoring method since it “anchors” on the smoothed prices when solving for the pseudo-dual prices. The current price estimate for item i in round t is therefore the pseudo-dual price, π it obtained by solving (QP). 5 3.3 Linear Pricing Algorithms Based on the Nucleolus

In this section, we provide alternative algorithms that better acknowledge that there will be bidders within the auction with both super-additive and sub-additive values as well as situations where the “bidders-are-substitutes” property does not hold. One compromise that attempts to mitigate between these alternative types of bidders and their respective 5 An alternative objective function to the quadratic objective function of (QP) is to minimize the maximum distance from the smoothed prices, subject to the same set of constraints. This alternative formulation would involve solving a sequence of minimization problems similar to the nucleolus approach that we will present later in this paper. The maximum number of linear programs that have to be solved is bounded by the number of licenses in the auction. This alternative formulation can overcome the numerical issues that could arise in large-scale problems when the objective function value based on the sum of squares becomes very large.

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bids is to provide linear pricing that, rather than treating the bidders as agents, treats “items” as the agents and tries to allocate the maximum revenue among these items “fairly”. Thus, we use the concept of a “nucleolus”. The nucleolus has been proposed as a method of allocating joint costs amongst multiple entities sharing a common resource.6 Here, we are sequentially minimizing the maximum dissatisfaction that could occur regarding a specific price estimate of a given item. The nucleolus algorithm considers the total costs incurred by various combinations (coalitions) of the entities in a cost allocation game. Here the costs are the prices of items that will be used to set acceptable bid prices for any package. Since no bidder is likely to accept bid prices that allocate the revenue in a way that is perceived to bias in favor of certain package prices over others, the analogy of “items as agents” in a cost allocation game, although not perfect, is one that has merit. For this reason, we argue for choosing prices that sequentially minimize the maximum deviation from “ideal” prices7. This is the idea behind a “nucleolus” solution. The nucleolus minimizes the potential dissatisfaction of any coalition with its share of the total cost savings, i.e. the difference between the costs savings that the coalition enjoys at the allocation, x and the cost savings that it could obtain by acting alone. The smaller this difference, the better off is the coalition at the allocation, x. For games where the core is empty, the algorithm minimizes the maximum dissatisfaction that any coalition can incur, fixes that value and continues to minimize the maximum dissatisfaction. When there are multiple allocations within the core, and joint action is better than unilateral action for every coalition, the algorithm works to assure that the excess obtained by this cooperation is shared equitably, i.e. it minimize the maximum amount that any coalition takes of the excess. Stated another way, the nucleolus is the set of allocations to which no group can validly object since it is the set of allocations, x with the property that for every objection there is a counter objection of equal value. An appealing property of the nucleolus is that there is always precisely one such allocation, i.e. it is unique. In our analogy, we are considering allocating the prices based on the bid prices provided in the auction so far. We would like these linear prices to (a) allocate the entire revenue, (b) be higher than the maximum “safe” bid price8 on the item or package, (c) satisfy the free-disposal property, (d) have the price of each package be no less than the high price bid on that package (dual feasibility) and (e) have the bid price on the winning packages equal the price of the winning bid amounts. When all of these properties cannot be satisfied, we will allow the bid prices to deviate from winning bid amounts, but force the 6

See Carter and Walker “The Nucleolus Strikes Back” Technical paper, Department of Economics, University of Canterbury and the citations provided therin for a complete discussion of the nucleolus and its game-theoretic properties. The concept of a nucleolus was first introduced by David Schmeidler (“The Nucleolus of a characteristic function game”. SIAM Journal of Applied Mathematics, 17 (1969) 11631170) For calculating the nucleolus, we use the algorithm of Kohlberg (“The Nucleolus as a Solution of a Minimization Problem” SIAM Journal of Applied Mathematics 21 (1972) 34-39) 7 We define “ideal” prices as prices that would satisfy all conditions of duality theory, while maintaining the linearity property. 8 A “safe” bid is defined to be the highest bid made on that package by a non-winning bidder. Thus, a “safe bid price” is the maximum price bid on that package by a non-winning bidder.

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sum of the item prices to equal the maximum revenue obtained by the winner determination problem in this round. To explain how the nucleolus calculation would work for this application, we begin by returning to our original winner-determination problem and its associated dual. If the linear optimization relaxation to the winner-determination problem was integer optimal than the resulting prices πi would satisfy the inequalities: ∑ π i ≥ b, ∀ j ∈ B and where the i∈L j

sum of the item prices would equal the revenue obtained in the winner determination problem. 3.3.1 The Nucleolus Algorithm

The first iteration of the nucleolus algorithm is: [ LP(δ 1 )] : min δ Subject to :

∑π

i

= MaxRevenue

∑π

i

+ δ ≥ bj ,

i∈L

(1) ∀ j∈B

(2)

π it ≥ ri ,

∀ i ∈I

δ

unrestricted in sign

( 3) ( 4)

i∈I j

In this problem, the constraints are similar to those used previously to obtain pseudo-dual prices. As before, we allow the prices of non-winning bids to be less than the maximum price bid for that package, but force the sum of the prices of items to equal the maximum revenue. However, we no longer require that δ be non-negative. Let δ1* be the solution to [ LP(δ 1 )] . Create the set J1* = { j |

∑π

i∈I

i

+ δ 1* = b j , j ∈ B} , and permanently fix δ = δ 1* , ∀j ∈ J1* .

j

Let J * = J1* . We can now solve the problem iteratively, where at iteration k, [ LP (δ k )] becomes:

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[ LP(δ k )] : min δ Subject to :

∑π

i

= MaxRevenue

∑π

i

+ δ ≥ bj ,

∀ j∈ B \ J*

(2)

∑π

i

= b j − δ *p ,

∀ j ∈ J *p , p = 1,..., k − 1

(3)

π i ≥ ri ,

∀i ∈ I

δ


( 4) ( 5)

i∈L

(1)

i∈I j

i∈I

j

J * = J1* ∪

... ∪ J k*−1

Let δ k* is the solution to [LP (δk)]. At the end of iteration k, we set J k* = { j |

∑π

i∈I

i

+ δ k* = b j , ∀ j ∈ B \ J *} , and permanently

j

fix δ = δ k* , ∀ j ∈ J k* . Set J * = J * ∪ J k* . The algorithm terminates when J * = B . The values πi after termination of the algorithm, is a unique set of item prices. We note that this algorithm may take many sequential steps, since the number of steps is dependent upon how many inequalities in (2) get fixed simultaneously. A worst-case instance may require the number of steps to be equal to the number of bids that are nonwinning. We can, however, reduce the number of steps by using a concept first proposed by Parkes (1999). Parkes suggests that the price of package should always be at least the price of a “safe” bid on that package. A “safe” bid is defined to be the highest bid made on that package by a non-winning bidder. Thus, we can preprocess the bids and for each non-winning package consider only the highest bid made on that package by a nonwinning bidder. For any winning package, we use the winning bid price. Thus, an upper bound to the number of iterations is reduced to the number of unique packages bid. We note that when the buyer submodularity property holds, the linear program solution will be integer optimal. Dual prices will exist such that there is no duality gap and the nucleolus algorithm will be choosing among prices in the core. 3.3.2 The Constrained Nucleolus Algorithm This algorithm is similar to the one discussed above, but we force the sum of the prices of items in a winning bid to equal the winning bid amount. Since this set of constraints is more constraining then the single revenue constraint, we have labeled this algorithm constrained nucleolus. The first iteration of the constrained nucleolus algorithm is:

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[ LP(δ 1 )] : min δ Subject to :

∑π

t i

= bj ,

∑π

i

+ δ ≥ bj ,

∀ j ∈W

(1)

∀ j ∈ B \W

(2)

π it ≥ ri ,

∀ i ∈I

δ


( 3) ( 4)

i∈I j

i∈I j

As before, we allow the prices of non-winning bids to be less than the maximum price bid for that package. Let δ1* be the solution to [ LP(δ 1 )] . Create the set J1* = { j |

∑π

i∈I

i

+ δ 1* = b j , j ∈ B \ W } , and permanently fix δ = δ 1* , ∀j ∈ J1* .

j

Let J = J . *

* 1

We can now solve the problem iteratively, where at iteration k, [ LP(δ k )] becomes: [ LP(δ k )] : min δ Subject to :

∑π

i

= bj ,

∀ j ∈W

(1)

∑π

i

+ δ ≥ bj ,

∀ j ∈ B \ (W ∪ J * )

(2)

∑π

i

= b j − δ *p ,

∀ j ∈ J *p , p = 1,..., k − 1

(3)

π i ≥ ri ,

∀i ∈ I

δ


(4) ( 5)

i∈I j

i∈I j

i∈I

j

J * = J 1* ∪

... ∪ J k*−1

Let δ k* is the solution to [LP (δk)]. At the end of iteration k, we set J k* = { j |

∑π

i∈I j

permanently fix δ = δ k* , ∀ j ∈ J k* . Set J * = J * ∪ J k* .

14

i

+ δ k* = b j , ∀ j ∈ B \ (W ∪ J * )} , and

The algorithm terminates when J * = B \ W . The constrained nucleolus algorithm is a more constrained version of the nucleolus algorithm, and has different convergence properties as will be demonstrated through simulations. 3.3.3 The RAD Algorithm We now present the pricing algorithm proposed by DeMartini, Kwasnica, Ledyard and Porter (1999). In their paper on “RAD” (Resource Allocation Design), they propose two alternative approaches for obtaining linear prices based on pseudo duals: (a) minimization of the maximum δj for all j∈B\W followed by maximization of the minimum πi for all i in item set L, in an iterative manner (b) minimization of the sum of the squares of δj for all j∈B \W. We note that the first of these two pricing schemes is very close to that of the nucleolus. The only difference is that the first stage of their algorithm stops as soon as the largest δj is no longer positive. At that point all other bids will satisfy dual feasibility. Once dual feasibility has been obtained, the algorithm maximizes the minimum price of any item. Specifically, the RAD algorithm’s first stage solves at iteration k the following problem (keeping notation as presented for the nucleolus above): [ LP(δ k )] :

δ k* = min δ Subject to :

∑π

i

= bj , ∀ j ∈W

(1)

∑π

i

+ δ j ≥ b j , ∀ j ∈ B \ (W ∪ J * )

(2)

∑π

i

= b j − δ p* , ∀ j ∈ J *p , p = 1,..., k − 1

(3)

i∈I j

i∈I

j

i∈I

j

π i ≥ ri , ∀ i ∈ I 0 ≤ δ j ≤ δ , ∀j ∈ B \ (W ∪ J * )

δ ≥0 J * = J 1* ∪

(4) ( 5) (6)

... ∪ J k*−1

This part of the RAD algorithm stops when the solution to problem LP(δ k ) has δ=0. RAD then reverts to a second sequential optimization procedure where one maximizes the minimum πi in a similar fashion. Namely, at iteration k of this sequence of optimizations, problem LP(π k ) has the following form, where the fixed π-values are placed into a set K*, comparable to the set J* for the fixed δ-values:

15

[ LP(π k )] :

π k* = max π Subject to :

∑π

i

= bj ,

∀ j ∈W

(1)

∑π

i

= b j − δ *p ,

∀ j ∈ J *p , p = 1,..., l

(2)

π i = π *p ,

∀ i ∈ K *p ,

(3)

π ≤ πi ,

∀ i ∈I \ K *

(4)

∀ i∈I

( 5)

i∈I j

i∈I

j

π i ≥ ri ,

p = 1,..., k − 1

K = K 1 ∪ ... ∪ K k −1 *

*

*

The algorithm stops when all prices are fixed, i.e. K* = L . Thus, in the first part of the algorithm, the RAD calculations are exactly the same as the constrained nucleolus, except that RAD considers bids on all packages rather than just the highest bid of a non-winning bidder on that package. Once dual feasibility is obtained, this algorithm reverts to trying to make the smallest item price as high as possible. In auctions where the various items have widely varying valuations, there is little justification to force prices of these items higher than the bidders have specified through their bidding information. The nucleolus, by contrast, continues to minimize the maximum deviation, δ even after the δ-values become non-positive. Thus, one continues to minimize the maximum difference between bid amounts and the associated price estimates, even after feasibility has been achieved. This seems especially reasonable to us if we first make the adjustment that we will only consider the high bids of nonwinning bidders for each package. All bids will remain in consideration for the winnerdetermination problem, but only winning bids on winning packages and high bids of nonwinning bidders on non-winning packages will determine the license prices. 3.3.4 The Smoothed Nucleolus (Nucleolus/Smoothed-Anchoring) Algorithm Finally, we now consider an alternative pricing algorithm that attempts to use some of the ideas of each of these alternatives: nucleolus, RAD, and smoothed anchoring coupled with safe prices. In the first step, we solve the nucleolus problem, but we stop, as RAD does, when dual feasibility is achieved. Specifically, we solve iteratively the problem LP(δ k ) for δ:

16

[LP (δ k )] : min δ Subject

to :

∑π

i

= bj , ∀j ∈W

(1)

∑π

i

+ δ j ≥ bj , ∀j ∈ B \ (W ∪ J * )

(2)

∑π

i

= bj − δ p*, ∀j ∈ J p*, p = 1,..., k − 1

(3)

i∈I j

i∈I

j

i∈I

j

∑

*

δ j = z δ* −

j ∈B \(W ∪J )

∑δ

j ∈J

* j

(4)

*

π i ≥ ri , ∀i ∈ I

(5)

0 ≤ δ j ≤ δ , ∀j ∈ B \ (W ∪ J * )

(6)

δ ≥0

(7 )

J * = J 1* ∪

... ∪ J k*−1

Constraint (4) imposes the duality gap constraint, where zδ* is the solution to the centering problem (CP). Let δ k* be the solution to [LP (δk)]. At the end of iteration k, we set J k* = { j |

∑π

i

+ δ k* = b j , ∀ j ∈ B \ (W ∪ J * )} , and permanently fix δ = δ k* , ∀ j ∈ J k* .

i∈I j

Set J * = J * ∪ J k* . We stop when δ k* = 0 . In the second step, we perform “smoothed anchoring”, by solving a quadratic (QP) problem:

[QP] : min

∑ (π i∈L

t i

− pit −1 ) 2

Subject to :

∑π

t i

≥ b j − δ pt* ,

∀ j ∈ J *p , p = 1,..., k

(1)

∑π

t i

≥ bj ,

∀ j ∈ B \ J*

(2)

∀ i ∈I

(3)

i∈I j

i∈I

j

π it ≥ ri ,

The model ensures prices obtained satisfy the dual feasibility conditions (1) and the free disposal constraints (2). The objective function minimizes the difference between the current round’s pseudo-dual price and the previous round’s smoothed price, pit −1 , which is 17

treated as a constant within the optimization, along with δ pt* , obtained from the series of iterations in the first step.

4

Simulation Framework and Test Design

To test the various linear pricing algorithms we used the intelligent-agent auction simulation tool BidBots. BidBots is composed of a collection of autonomous bidding agents each with a bidding strategy, a collection of items that it values (both separately and with synergistic interactions), a budget and a set of goals. The system has the ability to change many of the rules associated with a variety of auction designs, all of which are variations on auctions considered by the FCC. It uses a combinatorial-optimization solver to determine the provisional winners in each round, and the minimum acceptable bid prices for each package in the subsequent round. By using automated bidders, we are capable of running through many rounds of an auction quickly and evaluating how changes in auction rules and bidding strategies impact the overall speed, efficiency, and scalability of a variety of auction designs. We first provide the basics of the BidBots system and then summarize the test design. 4.1 Functions of Autonomous Bidding Agents

Each autonomous bidding agent, called a BidBot, is assigned values for some subset of the items being auctioned. These autonomous bidders then use this information to dynamically create new packages, alter the bid amounts on bids already placed in the auction, utilize activity waivers and/or reduce eligibility. Each bidder has a given budget that limits their bidding. Since this bidding system is designed to test FCC auction designs, the bidders use the information regarding the geographic license adjacency and bandwidth adjacency to form packages with synergistic values for geographic adjacency (complementary licenses) and also create packages within a region where alternative bandwidth frequencies can be substitutes. The bidders take advantage of the optimization software built into the system to determine the optimal bids to place (myopic-best response) based on their valuations and the current minimum acceptable bid prices. Bidders have the ability to perform jump bidding, and also to be straight-forward bidders moving from their first choice business plan to alternative business plans as the auction proceeds. There are three different classifications of bidders: 1. “National” bidders (These bidders want large portions of the geography of the US.) 2. “Regional” bidders (These bidders are interested in collecting bandwidth within a given geographic region, or regions.) 3. “Community” bidders (These bidders are only interested in smaller, local areas.) The user can determine the proportion of bidders that fit into each of these classifications. 4.2 Package Valuations by Bidding Agents

The user can specify the estimated auction revenue for a specific auction simulation in Bidbots. Based on this, the estimated market value of licenses l ∈ L is computed as follows:

18

lemv = (estimated auction revenue )(lbu / ∑ lbu ) l ∈L

Let Mb be the set of “home” markets for bidder b ∈ B . Then the valuation of a license by a bidding agent will be, b lval = {[0.9,1.1) − 0.05(min(hops(l , m )))}lemv , ∀ b ∈ B, l ∈ L m∈Mb

where, hops(l , m ) , is the number of hops license, l is away from the agent’s home market, m. The range [0.9, 1.1) is bidder type specific. Finally, the package valuations are computed as follows: b b pval = ( ∑ lval )[1.0, bsyn ), ∀ b ∈ B, pb ∈ P b l ∈pLb

where, bsyn is the maximum synergy and Pb be is the set of packages for bidder, b and pL is the set of licenses in package, p. We note that we have not modeled sub-additivity in this testing. We discuss the reasons for this omission. Most importantly, when the buyer submodularity property holds, the linear program solution will be integer optimal. Dual prices will exist such that there is no duality gap and all pricing algorithms will be choosing among prices in the core. Thus, we will naturally obtain dual prices that are consistent with theory when submodularity holds. When there are both sub-additive and super-additive valuations, then sub-additive packages are likely to be priced in an additive value because it is not possible for the pricing algorithm to distinguish among the individual “identical” items. Thus, each of the linear pricing algorithms will “average” the prices of identical items. We acknowledge this problem, but to correct this problem, one must alter the language of the bidding. That is, the language must allow preferences of the form “I will pay x for one unit of item A, x+ y for two units of A, x+y+z for three units, etc.” The auction we have modeled does not have this capability. Also, our current set of test problems is very small and we therefore are only simulating an auction within one band of spectrum. When we model larger auctions, we will have multiple bands being auctioned and it is these multiple bands that will provide the sub-additivity natural in our application, i.e. bidders are likely to treat multiple bands in a sub-additive fashion. Future testing will consider this case. 4.3 Auction Design

For this test effort the bidding agents in the BidBots system participate in an AusubelMilgrom Ascending Proxy Auction design using their package valuations. In this design, a proxy is assigned to each bidder. This proxy is given a set of packages that the bidder values positively and their respective bid amounts. In the pure Ausubel-Milgrom Ascending Proxy Auction, prices on all non-winning bids are incremented by a fixed amount, ε. The difference between the current minimum acceptable bids and the bidders’ values on each package determine the “profitability” of each package. Since only nonwinning bids are incremented, all winning bidders will not bid in the next round, since 19

their winning bid is “most profitable”. For non-winning bidders, the proxy determines the most profitable package and places a bid on that package. These bids are, in a sense, “myopic-best response” bids based on the local information of the bidder. If there are tied optimal bids, then multiple bids are sent to the system. All bids of a bidder are treated as mutually exclusive with all other bids placed by that bidder. All bids of all bidders are kept throughout the auction (i.e. there are no withdrawals), all bids of a given bidder are mutually exclusive (i.e. the bidder can win at most one of its submitted bids) and the auction has a soft stopping rule, i.e. the auction ends when there are no new bids in a round by any of the participating bidders. We note that this auction design uses nonlinear, non-anonymous pricing. We will first record the maximum revenue and the valuations of the winning bidders in the Ausubel-Miglrom auction implementation using alternative fixed increments. We will then record the corresponding auction results when the fixed small increments are replaced with linear price estimates and the minimum acceptable bid price is equal to the sum of the price estimates associated with the package plus the fixed increment. All other aspects of the auction design will remain as they are for the Ausubel-Milgrom auction with all bidders supplying only their myopic best-response bids to the auctioneer each round. For the linear price estimate auctions, we will also vary the increment and see how sensitive the results are to changes in the increment size. All simulations will assume private valuations, i.e. package valuations of each bidder are specified at the start of the auction and will not be revised during the course of the auction. We also will test our linear pricing against an accelerated proxy mechanism that has most of the properties of the proxy mechanism but solves to a greater accuracy in much less time and provides theoretically nice pricing properties when the “agents-are-substitutes” property holds, but the buyer submodularity does not hold. This mechanism is described in Section 4.6. 4.4 Simulation Design

The simulations were designed to highlight the relative performance of the ascending proxy design and the linear pricing schemes. The equilibrium outcomes implemented by each of these schemes depend on the satisfaction of certain conditions. The following definitions are important to understand the nature of a specific bargaining problem, and to make a qualitative statement of the allocations achieved in an auction. Free Disposal: Proxy mechanisms assume that the agent valuations satisfy the property of free disposal. Free disposal requires that for any agent, a, va (G ) ≥ va (T ), ∀G ⊇ T . This implies that any agent’s value on any superset of items, G, is at least as great as the agent’s value for a set T ⊆ G, i.e. there is no cost in receiving additional items at zero price. Competitive Equilibrium: An allocation, S, prices, p, and valuations v, are in competitive equilibrium if and only if the following two conditions hold:

20

va (Sa ) − pa (Sa ) = max( va (Sa' ) − pa (Sa ' )), ∀a ∈ A \ 0 '

∑

a∈A\0

S ∈X

pa (Sa ) = max ' S ∈X

∑

pa (Sa ' )

a∈A\0

where, A represents the set of participating agents, 0 represents the seller, Sa represents a’s package in the allocation S, and X denotes the space of feasible allocations. Therefore, (S, p) a competitive equilibrium if allocation S maximizes the payoffs for all agents including the seller, at the prices, p. Vickrey-Clark-Groves (VCG) Payoffs: The VCG payoff to agent a, svcg,a is computed as:

svcg ,a = w(A) − w(A \ a ), ∀a ∈ A* where, A is the set of all agents (including the seller who is denoted as 0), a∈A*, represents a winning bidder, the set of winning bidders is denoted as A*, and w(A) is the value of the winner determination problem when the valuations on the packages are used as the objective function coefficients in the winner determination problem. The seller payoff is computed as svcg ,o = w(A) − ∑ svcg ,a . The VCG payments by each bidder can a∈A*

be computed as va − svcg ,a , where va is the value that bidder a places on the item that he wins. For these VCG payoffs to have desired properties in a combinatorial auction, the “agents-are-substitutes” property must be satisfied. We define this property next. Agents-are-Substitutes (AAS): Based on bidder preferences, this is a condition whereby the following holds for all coalitions of buyers that do not include the seller:

w( A) − w( A \ K ) ≥ ∑ [ w( A) − w( A \ a)], ∀K ⊂ A, 0 ∉ K a∈K

The Agents-are-Substitutes (AAS) condition is a necessary condition for the VCG payments to be supported in the core, and for the existence of a unique buyer-optimal core payoff vector. However, this is not a sufficient condition for an ascending auction to terminate with VCG payments. For this to occur, the stronger condition of buyer submodularity must hold. Buyer Submodularity (BSM): The condition of buyer submodularity requires that the VCG payoff is in the core for all subcoalitions that include the seller: w( L) − w( L \ K ) ≥ ∑ [ w( L) − w( L \ a )], ∀K ⊂ L, 0 ∉ K , a∈K

for all L ⊆ A, 0 ∈ L where, L represents subcoalitions of all agents, A that include the seller. This stronger condition is sufficient for truthful, straightforward bidding and for an ascending proxy auction to terminate with the unique buyer Pareto-dominant Vickrey outcome.

21

Pareto-dominant allocation: An allocation, x′ is said to Pareto-dominate an allocation, x if every agent prefers x′ to x. A buyer Pareto-dominant allocation is one where every buyer prefers x′ to x. If the VCG outcome is in the core, then it is a Pareto-dominant allocation. Also, the core contains the Pareto-dominant point if and only if the VCG payment is in the core (Theorem 6, Ausubel-Milgrom [2001]). Failure of buyer submodularity condition could result in a buyer paying more than the VCG payment, by following a straightforward bidding strategy in an ascending proxy auction. We note, however, that our accelerated proxy auction will end with VCG payoffs whenever the “agents-are-substitutes” property holds, even when buyer submodularity may not. When the condition of “agents-are-substitutes” does not hold, the VCG payoff vector may not be in the core. In this case, truthful bidding is not an equilibrium strategy. Although the ascending proxy will converge to a core outcome, the total revenue obtained may be higher than a buyer optimal outcome, even when such an outcome exists. When no such unique outcome exists, Parkes (2001) has studied the incentive properties of allocations, and proposes a Threshold scheme for payment allocation that essentially minimizes the maximum deviation of payoffs from the VCG payoffs. The allocation achieved using this scheme is a “nucleolus” solution as defined in the previous sections, where the maximum surplus allocated to any bidder is minimized. Payment allocation using this scheme minimizes the bidder’s freedom to manipulate the outcome of the bargaining problem by misstating valuations. Threshold Scheme: According to this scheme payoff, πi, for all bidders, are computed as follows: min max[svcg ,a − sa ] π

a ∈A \0

s.t . so = w(A) −

∑

sa

a ∈A \0

∑s

a

≤ w(A) − w(A \ K ), ∀K ⊂ A, 0 ∉ K

a ∈K

Core Constraints

sa ≥ 0, ∀a ∈ A We will compare the outcomes achieved by linear pricing to those obtained by the proxy mechanism and this threshold scheme.

4.5 Test Cases We use two sets of test cases for our performance analysis. The first set of six small examples allows the reader to understand how pricing may be impacted when certain properties do not hold. The first three cases satisfy the agents-are-substitutes property,

22

while the last three do not. The buyer submodularity condition fails for Examples 2 and 3. The second problem set presented in the Appendix, consists of a set of ten profiles created using the BidBots simulation tool. We generated these instances based on a certain degree of randomness in the choice of package valuations and synergies by the bidding agents. Each instance consisted of ten bidders bidding on six heterogeneous items. The total estimated value of the auction was specified to be approximately $3 million. Packages comprising adjacent regions had synergistic valuations while packages containing non-adjacent regions had strictly additive valuations. We studied the impact of increment size, by running the same auctions for increments of $5,000, $30,000 and $60,000.

4.6 Measures We evaluate the results obtained from running auctions using linear pricing to those running a pure proxy auction described by Ausubel-Milgrom (2001). This proxy auction framework requires that all bidders provide to the auctioneer the maximum they are willing to pay for each item or package. The proxy then bids for the bidder using a myopic best-response strategy. The authors have shown that, for sufficiently small bid increments, the auction will always result in an efficient outcome with prices in the core and that when the buyer submodularity property is satisfied straightforward bidding is an equilibrium strategy, i.e. the auction is guaranteed to reach a core outcome with buyer Pareto-dominant prices. However, when the buyer submodularity condition does not hold, this property breaks down and sincere myopic best response bidding strategy is likely to lead a bidder to pay more than the optimal price for the winning item or package in the ascending proxy auction. To begin our testing, we first implemented the pure proxy auction and ran it for a given profile of six licenses, 10 bidders and valuations where the AAS (Agents-are-Substitutes) property does not hold. We used an increment size of $10, and the auction required over nine million rounds to converge. We realized that we could not use such a small increment and perform tests with auctions involving as small as 6 items and 10 bidders. In a separate research, described in “Accelerating the Ascending Proxy Auction” (2003), we have developed an alternative proxy mechanism, called the “Safe-Start IncrementScaling mechanism”. In this paper we will refer to this mechanism as the Accelerated Proxy mechanism. Unlike the pure proxy implementation, this Accelerated Proxy algorithm does not require the stronger condition of buyer submodularity to converge to VCG payments. Since the accelerated scheme starts by solving for the optimal allocation and is not sensitive to increment sizes, we use the allocation provided by this mechanism to benchmark each of the four linear pricing schemes, namely, (1) nucleolus approach (2) constrained nucleolus, (3) smoothed anchoring9 (4) smoothed nucleolus and (5) RAD. We therefore compare our results with (1) the Ausubel-Milgrom pure proxy mechanism using an increment of $5000 and (2) the accelerated proxy mechanism. We also provide the VCG payments for comparison.

9

Consistent with prior practice at the FCC, a weight factor of α=0.5 was used in this testing.

23

In both the small test set and the larger, but still quite small, simulations with 6 items and 10 bidders, we report the winning set, number of rounds required for completion and the average deviation of the payments based on linear pricing from the optimal bidder payments as determined by the accelerated proxy scheme. For the average price performance, we consider only profiles for which the pricing schemes produced the efficient outcome. The graphs showing the linear license price variations across rounds for each of the small examples are presented in the appendix, along with the set of plots for one of the larger profile 4.

4.7 Analysis We begin our analysis by discussing results based on the outcomes achieved for a few small test examples. These contrived cases serve to highlight specific kinds of bargaining problems when there are thin markets with little competition among packages. This can occur even in large auctions when there are few bidders interested in a given set of licenses. Example 1 of this set shows the simplest case, where the VCG payments are in the core, and all linear pricing schemes converge to the optimal allocation.

Example 1: Agents are Substitutes, buyer Submodularity satisfied Agent Package Value Method Pure Proxy Nucleolus Approach Constrained Nucleolus Smoothed Anchoring Smoothed Nucleolus RAD Approach Accelerated Proxy VCG

1 AB* 15

2 AB 14

C* 5

Rounds

Revenue

238 151 151 171 171 146 57 -

17.2 17.1 17.1 17.2 17.2 17.1 17 17

3 AB 9

4 AB 10

C 4

Payments By Winning Agents A1, {AB} A2, {C} 13.1 4.1 13.1 4.0 13.1 4.0 13.1 4.1 13.1 4.1 13.1 4.0 13 4 13 4

Although the final result is exactly what one desires, the plots (in the appendix of this report in Figure 3) illustrate that the individual price estimates across rounds are quite different for each of the four distinct pricing schemes discussed. Unlike RAD and the nucleolus approaches the algorithms that use smoothing produce near monotonic price estimates. The smoothed approaches weighted equally the prices of licenses A and B, while the nucleolus approach allocated the entire AB package price to license A. Since the bids in the auction provided no information about how to divide the cost between these licenses, it is not surprising that the allocations were quite different. We next examine Examples 2 and 3. Here agents are substitutes but the buyer submodularity property does not hold.

24

Example 2: Agents are Substitutes, buyer Submodularity not satisfied Agent Package Value

1

2

3

4

5

AB 21

BC 35

C 14

C* 20

AB* 22

Method Pure Proxy Nucleolus Approach Constrained Nucleolus Smoothed Anchoring Smoothed Nucleolus RAD Approach Accelerated Proxy VCG

Rounds

Revenue

403 297 351 298 298 291 6 -

36.9 35.2 35.1 35.05 35.05 35.02 35 35

Payments By Winning Agents A4, {C} A5, {AB} 15.8 21.1 14.1 21.1 14 21.1 13.99 21.06 13.99 21.06 14.03 20.99 14 21 14 21

Example 3: Agents are Substitutes, buyer Submodularity not satisfied Agent Package Value Method Pure Proxy Nucleolus Approach Constrained Nucleolus Smoothed Anchoring Smoothed Nucleolus RAD Approach Accelerated Proxy VCG

1 AB* 10

2 CD 20

3 CD* 25

Rounds

Revenue

326 202 184 202 202 202 35 -

27.7 26.9 25.9 30.1 30.1 26.9 20 20

4 BD 10

5 AC 10

Payments By Winning Agents A1, {AB} A3, {CD} 7.6 20.1 6.8 20.1 5.95 19.95 10 20.1 10 20.1 6.8 20.1 0 20 0 20

In Example 2, each of the linear pricing schemes results in prices closer to the VCG prices. However, the Ausubel-Milgrom auction produces higher prices, due to failure of the buyer submodularity property. All approaches except RAD eventually place all of the cost of package AB on package A. RAD averages the price between the two licenses. RAD has little fluctuation across rounds while the other approaches show a major adjustment at around round 141 (see Figure 4 in appendix). In Example 3, none of the linear pricing schemes converge to the VCG prices. Notice, however, that the Accelerated Proxy calculation does produce the VCG prices, which are buyer Pareto-dominant prices. Of the linear pricing approaches, the constrained nucleolus achieves results closest to the accelerated proxy and VCG prices, but shows significant fluctuations while attempting to adjust the prices. We now provide three examples where the “agents-are-substitutes” property is violated. The VCG prices are not supported in the core for these problems, and are an underestimate of prices in the core.

25

Example 4: Agents are not Substitutes Agent

1 A* 16

Package Value Method Pure Proxy Nucleolus Approach Constrained Nucleolus Smoothed Anchoring Smoothed Nucleolus RAD Approach Accelerated Proxy VCG Threshold

2 B 16

Rounds

Revenue

151 102 68 68 68 102 10 -

10.2 10.2 10.1 10.1 10.1 10.2 10 2 10

A 8

3 B* 8

AB 10

Payments By Winning Agents A1, {A} A2, {B} 5.1 5.1 5.1 5.1 5.05 5.05 5.05 5.05 5.05 5.05 5.1 5.1 5 5 2 0 6 4

Example 5: Agents are not Substitutes Agent Package Value Method Pure Proxy Nucleolus Approach Constrained Nucleolus Smoothed Anchoring Smoothed Nucleolus RAD Approach Accelerated Proxy VCG Threshold

1 AB* 15

2 C 5

BC 15

3 B 5

Rounds

Revenue

190 151 150 161 151 153 12 -

17.2 15.39 16.86 17.03 17.88 17.35 17 15 17

AC 12

C 3

4

5

AB 12

C* 6

Payments By Winning Agents A1, {AB} A5, {C} 12.1 5.1 12.05 3.34 11.91 4.95 13.45 3.58 11.96 5.92 12 5.35 13 4 12 3 13 4

Example 6: Agents are not Substitutes Agent Package Value Method Pure Proxy Nucleolus Approach Constrained Nucleolus Smoothed Anchoring Smoothed Nucleolus RAD Approach Accelerated Proxy VCG Threshold

1 AB 20

2 BC* 26

Rounds

Revenue

311 201 84 234 234 257 16 -

24.2 24 26 24.33 24.33 23.95 24 8 24

3 AC 24

4 A* 16

Payments By Winning Agents A2, {BC} A4, {A} 12.1 12.1 8.1 15.9 17.4 8.6 12.19 12.14 12.19 12.14 8.3 15.65 17 7 8 0 16 8

In Example 4, all pricing methods yielded results within one increment of the efficient allocation (where the increment in all of the small examples is 0.1). All prices converge

26

in a smooth manner. In Examples 5 and 6, each of the pricing algorithms converges to a distinct price allocation. In Example 5, the Accelerated Proxy and the Threshold scheme results in $17 revenue. Pure proxy, constrained nucleolus and smoothed anchoring algorithms come closest to this result, with nucleolus approach being farthest from the result and closest to the VCG underestimates. All methods require adjustments of the prices downward at some point in the auction. In Example 6, the constrained nucleolus allocation achieves a distribution of payments with weighting similar to those of the threshold scheme and the accelerated proxy. The pure proxy approach and the smoothed approaches produced prices with a more equal weighting. The nucleolus and RAD approaches, on the other hand, weight the payments heavily toward agent 4. Except for the constrained nucleolus, all approaches resulted in revenues close to $24 to the seller. Not surprisingly, the smoothing algorithms appear to handle new pricing information with fewer oscillations than does RAD. We next consider the results obtained from the ten profiles that constitute larger simulations involving 6 items and 10 bidders. A tabulation of the results from all these runs appears in the Appendix. This set consists of problems where synergies are the natural result of adjacent markets creating additional value to the buyer. In these instances, with an increment of $5000, for an auction whose total value was between $3.1 and $4.4 million dollars, the distribution of the number of items in the efficient outcome varied from one profile with a single global package, 4 profiles with 2 packages, 2 profiles with 3 packages and 3 profiles with 4 packages making up the efficient allocation. We summarize the results for each Profile in Table 1. We emphasize at the onset that the increment size was set at $5000. One must therefore consider a pricing mechanism to have succeeded if it obtained revenue within this tolerance. Based on this criterion, with very few exceptions, all methods successfully converged to within the revenue tolerance, regardless of whether or not the agents-are-substitutes property held. Interestingly, even when a pricing mechanism did not arrive at the efficient outcome, it did arrive at very close to the optimal revenue. One noted exception is Profile 3. This profile was constructed with synergy values that were very high. In this case, the agents-aresubstitutes property held and all methods obtained the efficient outcome. However, there was a far greater divergence in the revenue obtained from the various pricing algorithms.

27

Profile

Agents are Substitutes?

Efficient Result?

Proxy

RAD

Revenue within tolerance ($5000)?

YES NO YES NO NO NO YES YES YES YES

All methods RAD only All methods Nucleolus only All but pure proxy All but RAD and nucleolus All methods All methods RAD only None

YES YES YES YES YES YES $7,000 $13,000 $8,000 YES

YES YES $23,000 $7,000 YES $10,000 $6,000 YES YES YES

Nucleolus

Constrained Nucleolus

Smoothed Nucleolus

Smoothed Anchoring

YES YES $11,000 YES YES YES YES $7,000 YES YES

YES YES $15,000 YES YES YES YES YES $7,000 YES

YES YES $16,000 YES YES YES YES $8,000 YES $10,000

YES YES $13,000 YES $15,000 YES $8,000 YES YES YES

Table 1: Performance of the pricing schemes for each profile

Next, we show the computational effort required to compute various linear price estimates. In Figure 1, we show the number of iterations to solve the pricing problem, while in Figure 2 we show the number of rounds each method requires. The graphs in Figure 1 are based on the runs using Profile 4, but all profiles yield similar results. The Smoothed Anchoring takes 2 linear optimizations each round. All other linear approximations require the solution of a sequence of linear programs. Not surprisingly, the nucleolus-based approaches using only the safe bids require significantly less number of iterations than when using all bids. Similarly, the smoothed nucleolus approach requires less number of iterations than RAD, because the second stage of RAD is replaced by a single linear program identical to that used in the second stage of smoothed anchoring. Since the graphs for nucleolus and constrained nucleolus take about the same number of LPs to solve, we graph only the nucleolus approach with safe bids. 30 N u c le o lu s A p p ro a c h (A ll B id s ) 25 # of LPs Solved

1 2 3 4 5 6 7 8 9 10

Number of Winning Packages 1 2 4 3 2 2 2 3 4 4

20

N u c le o lu s A p p ro a c h (S a fe B id s )

15

R A D A p p ro a c h

10

S m o o th e d A n c h o rin g

5 S m o o th e d N u c le o lu s

0 1

21

41

61

81

101

Round

Figure 1: Computational effort required for each pricing algorithm

In Figure 2, we present the number of auction rounds using each of the linear pricing schemes. For Profile 4, the nucleolus approach took over 1500 rounds to converge to the efficient outcome. Except for that single outlier, all schemes showed comparable performance in terms of auction run times.

28

2000

Rounds

1500 1000 500 0 1

2

3

4

5

6

7

8

9

10

Profile

Nucleolus Approach

Constrained Nucleolus

Smoothed Anchoring

Smoothed Nucleolus

RAD

Figure 2: Auction durations using each linear pricing algorithm

In the appendix, we also provide results that show how the number of rounds and the revenue results change as the increment is changed from $5000 to $30,000 and $60,000 for each of the methods, for profiles 3, 4, and 9. Not surprisingly, the amount of time to complete the auctions reduces significantly as the increment size increases, but also with a greater deviation from second prices and a likely loss of efficiency. The degradation in performance of each of the linear pricing algorithms is mostly a function of the increment size, to which an ascending proxy design is equally susceptible. Finally, in Table 2, we present the average performance of each of the pricing schemes in terms of the average number of rounds to complete the auction, the mean and median of the absolute revenue deviations and absolute price deviations in the individual agent payments from that of the accelerated proxy. For the average price performance, we consider only profiles for which the pricing schemes produced the efficient outcome. Here, pure proxy achieved the efficient outcome for five of the ten profiles while each of the linear pricing schemes achieved the efficient outcome for six of the ten profiles. Method

Average Number of Rounds (Increment Size: $5,000)

Proxy Nucleolus Approach Constrained Nucleolus Smoothed Anchoring Smoothed Nucleolus RAD Approach

21,260 714 573 526 527 562

Abs. Revenue Deviation From Accelerated Proxy Revenue ($) Mean Median Max.

Abs. Price Deviation From Accelerated Proxy Price ($) Mean Median Max.

4,551 4,989 3,283 4,828 4,539 5,446

3,192 3,870 3,366 4,152 3,283 2,964

3,683 3,682 1,016 2,483 2,161 2,508

12,825 10,895 15,575 14,949 16,330 22,799

2,878 2,468 1,651 3,194 2,170 2,108

5,536 16,482 16,482 16,635 16,561 16,482

Table 2: Comparison of average performance of the pricing schemes

An analysis of the results shows that all of the linear pricing schemes perform relatively well, compared to the ascending proxy design implemented with nonlinear, nonanonymous pricing. There is a significant reduction in the average number of rounds for convergence to equilibrium outcomes, by using any of the linear pricing schemes. The nucleolus and RAD designs showed performance comparable to that of the smoothed 29

anchoring approach, but required more computational effort in terms of the number of linear programs that had to be solved in each round. We note that linear pricing schemes are likely to obtain better results as the number of agents increase. In this case, many agents may be bidding on only a few items and these small packages are likely to provide the necessary item-price information. Such auctions are also more likely to satisfy the “agents-are-substitutes” property. Our limited testing seems to indicate that one needs very high synergies (around 1.3 times the license prices), for any of the linear pricing algorithms to not converge to the optimal second prices. Here the pure ascending proxy mechanism that uses non-linear and non-anonymous prices out-performed the linear pricing (see Profile 3).

5

Based on this very limited testing, there is no clear-cut winner among the linear pricing schemes. The nucleolus approach seems to be deviating least from the second revenue prices, but requires the most number of computations. The prices estimated using this approach also has the theoretical property of being least objectionable to any bidder in a particular round. The smoothednucleolus seems to arrive at similar results with somewhat less computation and less price fluctuations. Future testing will examine if the same results hold when studying much larger auctions. Preliminary results seem to indicate that linear pricing performs quite well.Conclusions

In this report we have examined four linear pricing algorithms against a pure proxy design and compared it to the second prices (minimum price required to beat the competition) obtained by using an accelerated proxy method. We conclude that linear prices perform well relative to the package and bidder specific pricing implemented by the proxy mechanisms; but the size of the increment will influence how close these prices come to exact second prices. The smoothed anchoring method currently implemented by the FCC converges to revenue and bidder payments that are on average close to the optimal, in the ascending proxy auction format. The approach requires the least computational effort of all the approaches investigated in this report. The convergence properties, of course, depend on the choice of the smoothing parameter and the minimum opening license prices. When the minimum opening license prices may not reflect the true relative values of the licenses in a spectrum auction, the magnitude of the smoothing parameter has to be adjusted so the anchoring scheme learns quickly based on the current new bids in the system. When linear price estimates are required in an auction implementation where there are no prior linear prices to anchor to, the nucleolus approach is the best viable approach. Future research could investigate the numerical

30

Deleted: ¶

issues and evaluate the computational effort, the revenue generation and how close the outcomes are to the efficient outcome for larger test sets.

31

Appendix Set of Ten Auction Profiles: (6 Licenses, 10 Bidders, Increment Size= $5,000). We place the Accelerated Proxy results in bold since we compare all results to these more accurate calculations (accuracy to within $1.00). Profile 1 (Agents are Substitutes) Optimal Allocation {6-6057}; Optimal Value = $3,477,255; Optimal Revenue = $3,304,979 Method

Rounds

Pure Proxy Nucleolus Approach Constrained Nucleolus Smoothed Anchoring Smoothed Nucleolus RAD Approach Accelerated Proxy

21574 649 599 620 629 646 10

Allocation {Agent-Package} 6-6057 6-6057 6-6057 6-6057 6-6057 6-6057 6-6057

Prices ($) 3,300,000 3,306,856 3,305,121 3,303,433 3,306,482 3,303,931 3,304,979

Revenue ($) 3,300,000 3,306,856 3,305,121 3,303,433 3,306,482 3,303,931 3,304,979

Value ($) 3,477,255 3,477,255 3,477,255 3,477,255 3,477,255 3,477,255 3,477,255

Profile 2 (Agents are not Substitutes) Optimal Allocation {6-1005, 8-6041}; Optimal Value = $3,404,653; Optimal Revenue = $3,391,427 Method

Rounds


24023 636 555 471 447 647 45

Allocation {Agent-Package} {1-1005, 8-6041} {1-1005, 8-6041} {1-1005, 8-6041} {1-1005, 8-6041} {1-1005, 8-6041} {6-1005, 8-6041} {6-1005, 8-6041}

Prices ($) {530,000; 2,860,000} {530,212; 2,856,370} {533,356; 2,858,900} {533,465; 2,858,788} {530,332; 2,856,645} {534,167; 2,854,333} {534,643; 2,856,784}

Revenue ($) 3,390,000 3,386,582 3,392,256 3,392,253 3,386,977 3,388,500 3,391,427

Value ($) 3,404,338 3,404,338 3,404,338 3,404,338 3,404,338 3,404,653 3,404,653

Profile 3 (Agents are Substitutes) Optimal Allocation {1-6007, 4-1003, 6-1005, 8-1006}; Optimal Value = $3,822,237; Optimal Revenue = $3,481,228 Method

Rounds


18185 611 448 387 394 529 437

Allocation {Agent-Package} {1-6007, 4-1003, 6-1005, 8-1006} {1-6007, 4-1003, 6-1005, 8-1006} {1-6007, 4-1003, 6-1005, 8-1006} {1-6007, 4-1003, 6-1005, 8-1006} {1-6007, 4-1003, 6-1005, 8-1006} {1-6007, 4-1003, 6-1005, 8-1006} {1-6007, 4-1003, 6-1005, 8-1006}

Prices ($) {2,055,000; 490,000; 490,000; 450,000} {2,050,355; 505,000; 484,805; 451,963} {2,054,353; 505,000; 484,834; 452,616} {2,053,039; 505,153; 484,782; 451,957} {2,056,925; 505,079; 485,644; 449,910} {2,058,386; 505,000; 489,659; 450,982} {2,056,381; 488,518; 486,877; 449,452}

Revenue ($) 3,485,000 3,492,123 3,496,803 3,494,931 3,497,558 3,504,027 3,481,228

Value ($) 3,822,237 3,822,237 3,822,237 3,822,237 3,822,237 3,822,237 3,822,237

Profile 4 (Agents are not Substitutes) Optimal Allocation {1-1002, 2-6035, 6-6015}; Optimal Value = $3,168,750; Optimal Revenue = $3,085,136 Method

Rounds


18252 1529 823 663 685 397 514

Allocation {Agent-Package} {1-6013, 2-6035, 6-1003} {1-1002, 2-6035, 6-6015} {1-6013, 2-6035, 6-1003} {1-6013, 2-6035, 6-1003} {1-6013; 2-6035; 6-1003} {1-6013, 2-6035, 6-1003} {1-1002, 2-6035, 6-6015}

32

Prices ($) {1,035,000; 1,575,000; 475,000} {507,218; 1,559,288; 1,013,927} {1,027,070; 1,578,256; 479,003} {1,041,356; 1,570,331; 474,264} {1,026,179; 1,584,569; 474,596} {1,041,250; 1,575,000; 475,773} {505,316; 1,568,128; 1,011,692}

Revenue ($) 3,085,000 3,080,433 3,084,329 3,085,951 3,085,344 3,092,023 3,085,136

Value ($) 3,168,436 3,168,750 3,168,436 3,168,436 3,168,436 3,168,436 3,168,750

Profile 5 (Agents are not Substitutes) Optimal Allocation {4-6053, 6-1003}; Optimal Value =$3,451,175; Optimal Revenue = $3,398,594 Method

Rounds


22339 693 682 858 811 694 62


Prices ($) {2,920,000; 475,000} {2,919,585; 476,684} {2,927,783; 473,117} {2,908,348; 475,297} {2,923,677; 476,757} {2,922,221; 476,984} {2,919,442; 479,152}

Revenue ($) 3,395,000 3,396,269 3,400,900 3,383,645 3,400,434 3,399,205 3,398,594

Value ($) 3,448,925 3,451,175 3,451,175 3,451,175 3,451,175 3,451,175 3,451,175

Profile 6 (Agents are not Substitutes) Optimal Allocation {6-6049, 10-1004}; Optimal Value = $3,425,814; Optimal Revenue = $3,408,337 Method

Rounds


24026 800 682 580 589 718 46


Prices ($) {2,895,000; 510,000} {2,898,009; 507,855} {2,899,628; 508,300} {2,896,700; 507,954} {2,898,165; 509,380} {2,892,569; 505,949} {2,900,110; 508,227}

Revenue ($) 3,405,000 3,405,864 3,407,928 3,404,654 3,407,545 3,398,518 3,408,337

Value ($) 3,425,814 3,423,465 3,425,814 3,425,814 3,425,814 3,423,465 3,425,814

Profile 7 (Agents are Substitutes) Optimal Allocation {1-1006, 10-6026}; Optimal Value = $4,412,389; Optimal Revenue = $4,142,083 Method

Rounds


26260 700 747 593 620 725 3909


Prices ($) {690,000; 3,445,000} {693,321; 3,450,824} {690,000; 3,448,661} {690,000; 3,444,312} {690,000; 3,452,104} {690,000; 3,446,926} {694,451; 3,447,632}

Revenue ($) 4,135,000 4,144,145 4,138,661 4,134,312 4,142,104 4,136,926 4,142,083

Value ($) 4,412,389 4,412,389 4,412,389 4,412,389 4,412,389 4,412,389 4,412,389

Profile 8 (Agents are Substitutes) Optimal Allocation {1-1002, 3-1003, 6-6051}; Optimal Value = $3,312,071; Optimal Revenue = $3,222,175 Method

Rounds


20201 488 418 375 374 421 135

Allocation {Agent-Package} {1-1002, 3-1003, 6-6051} {1-1002, 3-1003, 6-6051} {1-1002, 3-1003, 6-6051} {1-1002, 3-1003, 6-6051} {1-1002, 3-1003, 6-6051} {1-1002, 3-1003, 6-6051} {1-1002, 3-1003, 6-6051}

33

Prices ($) {510,000; 475,000; 2,250,000} {507,833; 471,488; 2,250,215} {504,623; 473,279; 2,243,191} {506,532; 475,463; 2,243,145} {508,434; 472,445; 2,249,425} {505,813; 470,347; 2,243,927} {505,316; 472,395; 2,244,464}

Revenue ($) 3,235,000 3,229,536 3,221,093 3,225,140 3,230,304 3,220,087 3,222,175

Value ($) 3,312,071 3,312,071 3,312,071 3,312,071 3,312,071 3,312,071 3,312,071


Rounds


18442 536 377 324 326 380 71

Allocation {Agent-Package} {1-6050, 6-6002, 10-1002} {1-6050, 6-6002, 10-1002} {1-6050, 6-6002, 10-1002} {1-6050, 6-6002, 10-1002} {1-6050, 2-1002, 6-6002} {1-6050, 6-1003, 8-1001, 10-1002} {1-6050, 6-1003, 8-1001, 10-1002}

Prices ($) {1,640,000; 965,000; 500,000} {1,640,386; 963,268; 504,390} {1,639,489; 960,071; 505,101} {1,637,430; 962,787; 499,137} {1,633,700; 502,691; 963,444} {1,635,000; 478,102; 486,435; 499,398} {1,630,993; 477,164; 485,835; 503,361}

Revenue ($) 3,105,000 3,108,044 3,104,661 3,099,354 3,099,835 3,098,935 3,097,353

Value ($) 3,139,774 3,139,774 3,139,774 3,139,774 3,137,819 3,141,270 3,141,270


Rounds


19294 499 401 389 394 464 132

Allocation {Agent-Package} {1-1002, 3-6034, 6-6016} {1-6013, 3-6038, 8-1001} {1-6013, 3-6034, 6-6002} {1-6013, 3-6038, 8-1001} {1-1002, 3-6034, 6-6015, 8-1001} {1-6013, 3-6038, 8-1001} {1-1002, 3-6034, 6-6015, 8-1001}

34

Prices ($) {505,000; 1,025,000; 1,495,000} {1,041,223; 1,501,294; 485,849} {1,036,633; 1,024,454; 963,669} {1,039,814; 1,500,543; 485,328} {510,299; 1,028,634; 1,010,527; 485,877} {1,041,708; 1,500,365; 485,170} {505,316; 1,022,809; 1,011,692; 485,835}

Revenue ($) 3,025,000 3,028,366 3,024,756 3,025,685 3,035,337 3,027,243 3,025,706

Value ($) 3,084,570 3,083,503 3,084,255 3,083,503 3,084,255 3,084,255 3,086,066

Pure Proxy Performance

Profile

Increment Size Rounds Revenue Value ($)

1

2

3

4

5

6

7

8

9

10

5,000 30,000 60,000 5,000 30,000 60,000 5,000 30,000 60,000 5,000 30,000 60,000 5,000 30,000 60,000 5,000 30,000 60,000 5,000 30,000 60,000 5,000 30,000 60,000 5,000 30,000 60,000 5,000 30,000 60,000

21,574 3,567 1,778 24,023 4,007 1,971 18,185 3,026 1,433 18,252 2,996 1,483 22,339 3,806 1,893 24,026 3,931 1,983 26,260 4,272 2,220 20,201 3,267 1,602 18,442 3,080 1,497 19,294 3,109 1,574

($)

($)

3,300,000 3,300,000 3,300,000 3,390,000 3,390,000 3,360,000 3,485,000 3,480,000 3,480,000 3,085,000 3,090,000 3,060,000 3,395,000 3,390,000 3,360,000 3,405,000 3,390,000 3,360,000 4,135,000 4,110,000 4,140,000 3,235,000 3,180,000 3,180,000 3,105,000 3,090,000 3,060,000 3,025,000 2,970,000 3,000,000

3,477,255* 3,477,255* 3,477,255* 3,404,338* 3,391,427 3,391,427 3,822,237* 3,822,237* 3,798,161 3,168,436 3,168,750* 3,135,704 3,448,925 3,419,658 3,400,870 3,425,814* 3,425,814* 3,402,968 4,412,389* 4,412,389* 4,412,389* 3,312,071* 3,312,071* 3,264,016 3,139,774 3,101,558 3,101,558 3,084,570* 3,082,320 3,056,356

* indicates that Pure Proxy obtained the efficient outcome

35

Nucleolus Approach Performance Profile 3 (2nd Revenue of Efficient Outcome: $3,481,228; Value of Efficient Outcome: $3,822,237) Increment Size ($) 5,000 30,000 60,000

Rounds 611 132 50

Revenue ($) 3,492,123 3,458,486 3,447,663

Value of Outcome ($) 3,822,237* 3,779,614 3,776,075

Profile 4 (2nd Revenue of Efficient Outcome: $3,085,136; Value of Efficient Outcome: $3,168,750) Increment Size ($) 5,000 30,000 60,000

Rounds 1529 242 122

Revenue ($) 3,080,433 3,064,608 3,009,358

Value of Outcome ($) 3,168,750* 3,159,253 3,166,186


Rounds 536 103 73

Revenue ($) 3,108,044 3,092,084 3,090,666

Value of Outcome ($) 3,137,819 3,101,558 3,137,819

Constrained Nucleolus Performance Profile 3 (2nd Revenue of Efficient Outcome: $3,481,228; Value of Efficient Outcome: $3,822,237) Increment Size ($) 5,000 30,000 60,000

Rounds 448 94 55

Revenue ($) 3,496,803 3,528,800 3,482,750

Value of Outcome ($) 3,822,237* 3,822,237* 3,785,251


Rounds 823 184 99

Revenue ($) 3,084,329 3,077,332 3,058,339

Value of Outcome ($) 3,168,436 3,168,436 3,058,339


Rounds 377 102 88

Revenue ($) 3,104,661 3,119,361 3,042,844

36

Value of Outcome ($) 3,137,819 3,137,819 3,137,819

Smoothed Anchoring Performance Profile 3 (2nd Revenue of Efficient Outcome: $3,481,228; Value of Efficient Outcome: $3,822,237) Increment Size ($) 5,000 30,000 60,000

Rounds 387 80 41

Revenue ($) 3,494,931 3,517,156 3,443,747

Value of Outcome ($) 3,822,237* 3,822,237* 3,761,028


Rounds 663 122 56

Revenue ($) 3,085,951 3,066,904 3,001,224

Value of Outcome ($) 3,166,186 3,168,436 3,138,287


Rounds 324 61 32

Revenue ($) 3,099,354 3,067,660 2,997,443

Value of Outcome ($) 3,141,270* 3,137,819 3,084,471

Smoothed Nucleolus Performance Profile 3 (2nd Revenue of Efficient Outcome: $3,481,228; Value of Efficient Outcome: $3,822,237) Increment Size ($) 5,000 30,000 60,000

Rounds 394 77 42

Revenue ($) 3,497,558 3,513,237 3,436,116

Value of Outcome ($) 3,822,237* 3,776,075 3,755,538


Rounds 685 124 68

Revenue ($) 3,085,344 3,123,352 3,105,844

Value of Outcome ($) 3,168,436 3,168,436 3,159,253


Rounds 326 57 42

Revenue ($) 3,099,835 3,081,681 3,076,795

37

Value of Outcome ($) 3,137,819 3,132,296 3,093,279

RAD Performance Profile 3 (2nd Revenue of Efficient Outcome: $3,481,228; Value of Efficient Outcome: $3,822,237) Increment Size ($) 5,000 30,000 60,000

Rounds 529 91 42

Revenue ($) 3,504,027 3,468,320 3,435,000

Value of Outcome ($) 3,822,237* 3,822,237* 3,742,329


Rounds 397 122 50

Revenue ($) 3,092,023 3,055,833 3,020,000

Value of Outcome ($) 3,168,436 3,168,436 3,166,186


Rounds 380 111 57

Revenue ($) 3,098,935 3,090,278 3,083,858

Value of Outcome ($) 3,139,021 3,137,819 3,095,529

* indicates the pricing approach obtained the efficient outcome

38

Nucleolus Approach License A

License B

14

1

12

0.8

License C 4.5 4 3.5

10 3

0.6 8

2.5 0.4 2

6

1.5

0.2 4

1 2

0

0

-0.2

0.5

1

11

21

31

41

51

61

71

81

91

0

101 111 121 131 141

1

11

21

31

41

51

61

Round

71

81

91

1

101 111 121 131 141

9 17 25 33 41 49 57 65 73 81 89 97 105 113 121 129 137 145 Round

Round

Constrained Nucleolus License A

License B

14

1

12

0.8

License C 4.5 4 3.5

10 0.6

3

8

2.5 0.4 2

6 0.2

1.5

4 1 2

0

0

-0.2

0.5

1

11

21

31

41

51

61

71

81

91

0

101 111 121 131 141

1

11

21

31

41

51

Round

61

71

81

91

101 111 121 131 141

1

9 17 25 33 41 49 57 65 73 81 89 97 105 113 121 129 137 145

Round

Round

Smoothed Anchoring License A

License B

7

7

6

6

5

5

4

4

3

3

2

2

1

1

0

0

License C 4.5 4 3.5 3 2.5 2 1.5 1

1

11

21

31

41 51

61

71

81

91 101 111 121 131 141 151 161

0.5 0 1

11

21

31 41

51

61

Round

71 81

1

91 101 111 121 131 141 151 161

11 21

31 41

51 61

71 81 91 101 111 121 131 141 151 161 Round

Round

Smoothed Nucleolus License A

License B

7

7

6

6

5

5

4

4

3

3

2

2

1

1

0

0

License C 4.5 4 3.5 3 2.5 2 1.5 1

1

11

21

31

41 51

61

71

81

91 101 111 121 131 141 151 161

0.5 0 1

Round

11

21

31 41

51

61

71 81

91 101 111 121 131 141 151 161

1

11 21

31 41

51 61

71 81 91 101 111 121 131 141 151 161

Round

Round

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License B

101 111 121 131 141

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Figure 3: Price variations in Example 1

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Nucleolus Approach License A 25

License B 12

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Smoothed Anchoring License A 25

License B

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Smoothed Nucleolus License A 25

License B

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RAD License A

License B

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License B

License C

License D

8

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License B

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License D

4

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License B

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License B

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License B

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License C

7

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License B

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License B

License C

7

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License B

License C

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RAD License A

License B

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License B

License C

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License B

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RAD License A

License B

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License 1

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License 3 500000 450000 400000

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License 5

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License 6

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Figure 9: Nucleolus price variations for Profile 4

License 1

License 2

License 3

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Figure 10: Constrained nucleolus price variations for Profile 4

45

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License 1

License 2

License 3

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Figure 11: Smoothed anchoring price variations for Profile 4

License 1

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Figure 12: Smoothed nucleolus price variations for Profile 4

46

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License 1

License 3

License 2

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Figure 13: RAD price variations for Profile 4

47

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Bibliography 1. Bikhchandani, S. and J.M. Ostroy (2001), “The Package Assignment Model”, UCLA Anderson School of Management Technical Report. 2. Bikchandani, S., S. DeVries, J. Schummer and R. Vohra (2001), “Linear Programming and Vickrey Auctions”, UCLA Anderson School of Management Technical Report. 3. Carter, M. and P. Walker, “The Nucleolus Strikes Back”, Technical paper, Department of Economics, University of Canterbury. 4. DeMartini, C., A.M. Kwasnica, J. Ledyard and D. Porter (1998), “A New and Improved Design for Multi-Object Iterative Auctions”, Social Working Paper, Division of the Humanities and Social Sciences, California Institute of Technology, Pasadena, CA. 5. DeVries, S. and R. Vohra (2003), “Combinatorial Auctions: A Survey”, INFORMS Journal on Computing Volume 15, No. 3 (Summer 2003). 6. Fujishima, Y., K. Leyton-Brown, and Y. Shoham (1999), “Taming the Computational Complexity of Combinatorial Auctions: Optimal and Approximate Approaches”, Technical Report, #F30602-98-C-0214. 7. FCC PN DA 02-260, February 4, 2002, FCC Auction 31 Public Notices, FCC Auctions website. 8. Hoffman, K., D. Menon, and T. Wilson (2003), “Observations and Near-Direct Implementations of the Ascending Proxy Auction”, Technical Report, George Mason University, Systems Engineering and Operations Research Department. 9. Kohlberg, E. (1972), “The Nucleolus as a Solution of a Minimization Problem”, SIAM Journal of Applied Mathematics 21 34-39. 10. Owen, G. (1995), Game Theory Academic Press, 322-334. 11. Parkes, D. C. and L.H. Ungar (2000), “Iterative Combinatorial Auctions: Theory and Practice”, Proc. 17th National Conference on Artificial Intelligence (AAAI-00) 74-81. 12. Rassenti, S., V. Smith and R.L. Bulfin (1982), “A Combinatorial Auction Mechanism for Airport Slot Allocation” Bell Journal of Economics 13 402-417. 13. Schmeidler, D. (1969), “The Nucleolus of a Characteristic Function Game”, SIAM Journal of Applied Mathematics, 17, 1163-1170. 14. Wolsey, L.A. (1981), “Integer Programming Duality: Price Functions and Sensitivity Analysis”, Mathematical Programming, 20, 173-195. 15. Wurman, P. R. and M.P. Wellman (1999), “Equilibrium Prices in Bundle Auctions”, Sante Fe Institute Working Papers, Paper #: 99-09-064.

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